Theorem univ 4454

Description: The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)

Assertion

Ref	Expression
univ	⊢ ∪ V = V

Proof of Theorem univ

Step	Hyp	Ref	Expression
1		pwv 3788	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 3799	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 4195	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2188	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1343  Vcvv 2726  𝒫 cpw 3559  ∪ cuni 3789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-uni 3790

This theorem is referenced by: (None)